The topological vs the smooth categories.
We have chosen to work in the differentiable category in order to present our results in a unified manner. However, if \(M\)  has dimension \(\le 3\) , then by [1]}, [2]}, [3]}
\(\pi _0(\operatorname{Homeo}^+(M,\partial M)) \cong \pi _0(\operatorname{Diff}^+(M,\partial M)),\) 
